Simplify the following expression and state the condition under which the simplification is valid: $k = \dfrac{r^2 - r - 20}{r^2 - 5r}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{r^2 - r - 20}{r^2 - 5r} = \dfrac{(r + 4)(r - 5)}{(r)(r - 5)} $ Notice that the term $(r - 5)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(r - 5)$ gives: $k = \dfrac{r + 4}{r}$ Since we divided by $(r - 5)$, $r \neq 5$. $k = \dfrac{r + 4}{r}; \space r \neq 5$